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A generalization of the [[Fundamental group|fundamental group]], proposed by W. Hurewicz [[#References|[1]]] in the context of problems on the classification of continuous mappings. Homotopy groups are defined for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479301.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479302.png" /> the homotopy group is identical with the fundamental group. The definition of homotopy groups is not constructive and for this reason their computation is a difficult task, general methods for which were developed only in the 1950s. Their importance is due to the fact that all problems in homotopy theory can be reduced (cf. [[Homotopy type|Homotopy type]]), to a greater or lesser extent, to the computation of certain homotopy groups.
+
{{TEX|done}}
 +
A generalization of the [[Fundamental group|fundamental group]], proposed by W. Hurewicz [[#References|[1]]] in the context of problems on the classification of continuous mappings. Homotopy groups are defined for any $  n \geq 1 $ .  
 +
For $  n = 1 $
 +
the homotopy group is identical with the fundamental group. The definition of homotopy groups is not constructive and for this reason their computation is a difficult task, general methods for which were developed only in the 1950s. Their importance is due to the fact that all problems in homotopy theory can be reduced (cf. [[Homotopy type|Homotopy type]]), to a greater or lesser extent, to the computation of certain homotopy groups.
  
Let
+
Let$$
 +
I ^{n}  =  \{ {( t _{1} \dots t _{n} )} : {
 +
0 \leq t _{1} \leq 1 \dots 0 \leq t _{n} \leq 1} \}
 +
$$
 +
be the dimensional unit cube, let $  I _{n-1} $
 +
be its face $  t _{n} = 0 $ ,
 +
and let $  J ^{n-1} $
 +
be the union of its remaining faces. For any pointed pair $  ( X ,\  A ,\  x _{0} ) $ (
 +
cf. [[Pointed object|Pointed object]]) the symbol $  \pi _{n} ( X ,\  A ,\  x _{0} ) $ (
 +
or simply $  \pi _{n} ( X ,\  A ) $ )
 +
denotes the pointed set of all homotopy classes (cf. [[Homotopy|Homotopy]]) of mappings$$
 +
u : \  ( I ^{n} ,\  I ^{n-1} ,\  J ^{n-1} )  \rightarrow  ( X ,\  A ,\  x _{0} ) .
 +
$$
 +
The distinguished (zero) element of this set is the constant mapping that maps the whole cube $  I ^{n} $
 +
into $  x _{0} $ .
 +
Any continuous mapping$$
 +
f : \  ( X ,\  A ,\  x _{0} )  \rightarrow 
 +
( Y ,\  B ,\  y _{0} )
 +
$$
 +
induces a [[Morphism|morphism]]$$
 +
f _ \star  : \  \pi _{n} ( X ,\  A ,\  x _{0} ) 
 +
\rightarrow  \pi _{n} ( Y ,\  B ,\  y _{0} )
 +
$$
 +
of pointed sets. For any $  n \geq 1 $
 +
the sets $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
and the morphisms $  f _ \star  $
 +
constitute a [[Functor|functor]] $  \pi _{n} $
 +
from the [[Category|category]] of pointed pairs into the category of pointed sets. This functor is homotopy invariant, i.e. $  f _ \star  = g _ \star  $
 +
if $  f $
 +
and $  g $
 +
are homotopic (as mappings of pointed pairs). Furthermore, it is normalized in the sense that if $  X = A = x _{0} $ ,
 +
then $  \pi _{n} ( X ,\  A ,\  x _{0} ) = 0 $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479303.png" /></td> </tr></table>
 
  
be the dimensional unit cube, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479304.png" /> be its face <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479305.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479306.png" /> be the union of its remaining faces. For any pointed pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479307.png" /> (cf. [[Pointed object|Pointed object]]) the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479308.png" /> (or simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h0479309.png" />) denotes the pointed set of all homotopy classes (cf. [[Homotopy|Homotopy]]) of mappings
+
For $  n \geq 2 $
 +
it is possible to introduce into the set $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
an operation of addition, with respect to which it becomes a group (if $  n \geq 2 $
 +
even an Abelian group). By definition, if $  x = [ u ] $
 +
and $  y = [ v ] $ ,
 +
then $  x + y = [ w ] $ ,
 +
where $  w $
 +
is the mapping$$
 +
( I ^{n} ,\  I ^{n-1} ,\  J ^{n-1} ) 
 +
\rightarrow  ( X ,\  A ,\  x _{0} )
 +
$$
 +
defined by the formula$$ \tag{1}
 +
w ( t _{1} \dots t _{n} )  =
 +
$$
 +
$$
 +
=
 +
\left \{
 +
\begin{array}{ll}
 +
u ( 2 t _{1} ,\  t _{2} \dots t _{n} )  &  \textrm{ if }  0 \leq t _{1} \leq 1 / 2 ,  \\
 +
v ( 2 t _{1} -1 ,\  t _{2} \dots t _{n} )  &  \textrm{ if }  1 / 2 \leq t _{1} \leq 1 . \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793010.png" /></td> </tr></table>
+
\right .$$
 +
The resulting group $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
is said to be the $  n $ -
 +
th homotopy group (or the $  n $ -
 +
dimensional homotopy group) of the pointed pair $  ( X ,\  A ,\  x _{0} ) $ ;  
 +
one also speaks of the homotopy group of the pair $  ( X ,\  A ) $
 +
at $  x _{0} $
 +
or of the homotopy group of the space $  X $
 +
with respect to the subspace $  A $
 +
at $  x _{0} $ .  
 +
The mappings $  f _ \star  $
 +
are homomorphisms of these groups. Thus, if $  n \geq 2 $
 +
it may be assumed that the function $  \pi _{n} $
 +
takes values in the category of groups (if $  n > 2 $
 +
even in the category of Abelian groups).
  
The distinguished (zero) element of this set is the constant mapping that maps the whole cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793011.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793012.png" />. Any continuous mapping
+
For $  A = x _{0} $
 +
the group $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
is denoted by $  \pi _{n} ( X ,\  x _{0} ) $ ,
 +
or simply by $  \pi _{n} (X) $ ,
 +
and is called the absolute homotopy group of the pointed space $  ( X ,\  x _{0} ) $ (
 +
or of the space $  X $
 +
at $  x _{0} $ ).
 +
Its elements are the homotopy classes of mappings $  ( I ^{n} ,\  \dot{I}  ^{n} ) \rightarrow ( X ,\  x _{0} ) $ ,
 +
where $  \dot{I}  ^{n} = I ^{n-1} \cup J ^{n-1} $
 +
is the boundary of the cube $  I ^{n} $ .
 +
For such mappings formula (1) is meaningful for $  n = 1 $
 +
as well, and so $  \pi _{n} ( X ,\  x _{0} ) $
 +
is a group. This group coincides with the classical fundamental group. The group operation in $  \pi _{n} ( X ,\  x _{0} ) $
 +
is usually called multiplication. This group is, generally speaking, non-Abelian, while the group $  \pi _{2} ( X ,\  x _{0} ) $
 +
is Abelian. For any $  n \geq 1 $
 +
the groups $  \pi _{n} ( X ,\  x _{0} ) $
 +
and the corresponding homomorphisms form a functor from the category of pointed spaces into the category of groups (if $  n > 1 $
 +
into the category of Abelian groups). This functor is the composition $  \pi _{n} \circ \iota $
 +
of the imbedding functor $  \iota : \  ( X ,\  x _{0} ) \rightarrow ( X ,\  x _{0} ,\  x _{0} ) $
 +
and the functor $  \pi _{n} $
 +
described above.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793013.png" /></td> </tr></table>
+
The functor $  \pi _{n} \circ \iota $
 +
is extended to include the case $  n = 0 $ ,
 +
where $  \pi _{0} ( X ,\  x _{0} ) $
 +
is the pointed set of path-components (cf. [[Path-connected space|Path-connected space]]) of $  X $ ;  
 +
the zero of this set is the component containing $  x _{0} $ .  
 +
The set $  \pi _{0} ( X ,\  A ,\  x _{0} ) $
 +
is not defined for $  A \neq x _{0} $ .  
 +
In order to simplify the formulations, the sets $  \pi _{0} ( X ,\  x _{0} ) $
 +
and $  \pi _{1} ( X ,\  A ,\  x _{0} ) $
 +
are usually also called homotopy groups, even though they are not groups in general.
  
induces a [[Morphism|morphism]]
+
For each element $  x = [ u ] \in \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
the mapping $  u \mid _ {I ^{n-1}} $
 +
represents a mapping $  ( I ^{n-1} ,\  \dot{I}  ^{n-1} ) \rightarrow ( A ,\  x _{0} ) $ ,
 +
and thus defines a certain element of the homotopy group $  \pi _{n-1} ( A ,\  x _{0} ) $ .
 +
This element depends only on $  x $
 +
and is denoted by the symbol $  \partial x $ .
 +
The resulting mapping $  \partial : \  \pi _{n} ( X ,\  A ,\  x _{0} ) \rightarrow \pi _{n-1} ( A ,\  x _{0} ) $
 +
is a morphism of pointed sets (if $  n > 1 $
 +
a homomorphism of groups) and is called a boundary homomorphism or a boundary operator. The boundary homomorphism, together with the homomorphisms $  i _ \star  $
 +
and $  j _ \star  $
 +
induced by the imbeddings $  i : \  ( A ,\  x _{0} ) \rightarrow ( X ,\  x _{0} ) $
 +
and $  j : \  ( X ,\  x _{0} ) \rightarrow ( X ,\  A ,\  x _{0} ) $ ,
 +
makes it possible to write down a sequence of groups and homomorphisms, infinite from the left:$$
 +
{} \dots  \stackrel{ {j _{\#}}} \rightarrow    \pi _{n+1}
 +
( X ,\  A ,\  x _{0} )    \stackrel \partial  \rightarrow    \pi _{n}
 +
( A ,\  x _{0} )    \stackrel{ {i _{\#}}} \rightarrow   
 +
\pi _{n} ( X ,\  x _{0} )    \stackrel{ {j _{\#}}} \rightarrow 
 +
$$
 +
$$
 +
\stackrel{ {j _{\#}}} \rightarrow    \pi _{n} ( A ,\  x _{0} )    \stackrel \partial  \rightarrow  \dots
 +
$$
 +
$$
 +
{} \dots  \stackrel{ {j _{\#}}} \rightarrow    \pi _{2} ( X ,\  A ,\  x _{0} )    \stackrel \partial  \rightarrow    \pi _{1} ( A ,\  x _{0} )    \stackrel{ {i _{\#}}} \rightarrow 
 +
  \pi _{1} ( X ,\  x _{0} )    \stackrel{ {j _{\#}}} \rightarrow   
 +
$$
 +
$$
 +
  \stackrel{ {j _{\#}}} \rightarrow    \pi _{1} ( X ,\  A ,\  x _{0} )    \stackrel \partial  \rightarrow    \pi _{0} ( A ,\  x _{0} )    \stackrel{ {i _{\#}}} \rightarrow    \pi _{0} ( X ,\  x _{0} ) .
 +
$$
 +
This is an [[Exact sequence|exact sequence]]; it is called the exact homotopy sequence of the pair $  ( X ,\  A ,\  x _{0} ) $
 +
and is usually denoted by $  \pi ( X ,\  A ,\  x _{0} ) $ .
 +
If $  \pi _{n} ( X ,\  x _{0} ) = 0 $
 +
for all $  n \geq 0 $ ,
 +
then the homomorphism $  \partial : \  \pi _{n} ( X ,\  A ,\  x _{0} ) \rightarrow \pi _{n-1} ( A ,\  x _{0} ) $
 +
is an isomorphism (also for all $  n $ ).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793014.png" /></td> </tr></table>
 
  
of pointed sets. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793015.png" /> the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793016.png" /> and the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793017.png" /> constitute a [[Functor|functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793018.png" /> from the [[Category|category]] of pointed pairs into the category of pointed sets. This functor is homotopy invariant, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793019.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793021.png" /> are homotopic (as mappings of pointed pairs). Furthermore, it is normalized in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793023.png" />.
+
The boundary homomorphism $  \partial $
 +
is natural, that is, it is a morphism of the functor $  \pi _{n} $
 +
into the functor $  \pi _{n-1} \circ \iota $ (
 +
more exactly, into the functor $  \pi _{n-1} \circ \iota ^ \prime  $
 +
where $  \iota ^ \prime  : \  ( X ,\  A ,\  x _{0} ) \mapsto ( A ,\  x _{0} ,\  x _{0} ) $ ).  
 +
This makes it possible to define $  \pi ( X ,\  A ,\  x _{0} ) $
 +
as a functor that takes values in the category of exact sequences of pointed sets which, except for the last six sets, are Abelian groups and, except for the last three sets, are groups.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793024.png" /> it is possible to introduce into the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793025.png" /> an operation of addition, with respect to which it becomes a group (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793026.png" /> even an Abelian group). By definition, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793030.png" /> is the mapping
+
Let $  p : \  E \rightarrow B $
 +
be an arbitrary [[Fibration|fibration]] in the sense of Serre and let $  A \subset B $ ,
 +
$  E ^ \prime  = p ^{-1} $ ,
 +
$  e _{0} \in E ^ \prime  $ ,
 +
and $  b _{0} = p ( e _{0} ) $ .
 +
The mapping $  p $
 +
defines a mapping $  p ^ \prime  : \  ( E ,\  E ^ \prime  ,\  e _{0} ) \rightarrow ( B ,\  A ,\  b _{0} ) $
 +
of pointed pairs. For any $  n \geq 1 $
 +
the induced homomorphism $  p _ \star  ^ \prime  : \  \pi _{n} ( E ,\  E ^ \prime  ,\  e _{0} ) \rightarrow \pi _{n} ( B ,\  A ,\  b _{0} ) $
 +
is an isomorphism. In particular, this is true for $  A = b _{0} $ .  
 +
In the latter case the formula $  \tau = \partial \circ ( p _ \star  ^ \prime  ) ^{-1} $
 +
unambiguously defines a homomorphism $  \tau : \  \pi _{n} ( B ,\  b _{0} ) \rightarrow \pi _{n-1} ( F ,\  e _{0} ) $
 +
where $  F = p ^{-1} ( b _{0} ) $
 +
is the fibre of $  p $
 +
over $  b _{0} $ .  
 +
This homomorphisms is called the homotopy transgression. It occurs in the exact sequence$$
 +
{} \dots \rightarrow  \pi _{n} ( F ,\  e _{0} ) 
 +
\stackrel{ {i _{\#}}} \rightarrow    \pi _{n}
 +
( E ,\  e _{0} )    \stackrel{ {p _{\#}}} \rightarrow   
 +
\pi _{n} ( B ,\  b _{0} )    \stackrel \tau  \rightarrow 
 +
$$
 +
$$
 +
\stackrel \tau  \rightarrow    \pi _{n-1} ( F ,\  e _{0} )  \rightarrow \dots .
 +
$$
 +
This sequence is called the homotopy sequence of the fibration $  p : \  E \rightarrow B $ .
 +
Putting a fibration into correspondence with its homotopy sequence yields a functor on the category of all (pointed) fibrations.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793031.png" /></td> </tr></table>
+
In the particular case when $  p $
 +
is the standard [[Serre fibration|Serre fibration]] of paths over a space $  X $ ,
 +
for any $  n \geq 0 $
 +
one has the isomorphism $  \pi _{n} ( \Omega X ) \approx \pi _{n+1} (X) $ ,
 +
where $  \Omega X $
 +
is the [[Loop space|loop space]] of $  X $ .  
 +
This isomorphism is called the Hurewicz isomorphism.
  
defined by the formula
+
The above properties actually unambiguously define the homotopy groups $  \pi _{n} ( X ,\  A ,\  x _{0} ) $ ,
 +
i.e. may be taken as axioms which describe these groups. In fact, let $  \pi _{1} \dots \pi _{n} \dots $
 +
be an arbitrary sequence of homotopy-invariant normalized functors, defined on the category of pointed spaces, taking values in the category of pointed sets, and having the following property: For any fibration in the sense of Serre $  p : \  E \rightarrow B $ ,
 +
any subset $  A \subset B $
 +
and any point $  e _{0} \in p ^{-1} (A) $ ,
 +
the induced homomorphism $  \pi _{n} ( E ,\  p ^{-1} (A) ,\  e _{0} ) \rightarrow \pi _{n} ( B ,\  A ,\  p ( e _{0} ) ) $
 +
is an isomorphism. Such a sequence is called a homotopy system if for any $  n \geq 1 $
 +
there is defined a morphism $  \partial $
 +
of the functor $  \pi _{n} $
 +
into the functor $  \pi _{n-1} \circ \iota ^ \prime  $ (
 +
if $  n = 1 $ ,
 +
into $  \pi _{0} ( X ,\  x _{0} ) $ )
 +
that is an isomorphism for any pointed pair $  ( X ,\  A ,\  x _{0} ) $
 +
for which $  \pi _{n} (X ,\  x _{0} ) = 0 $
 +
for all $  n \geq 0 $ .
 +
Any homotopy system is isomorphic to the homotopy system constructed above, which consists of homotopy groups. Furthermore, if $  n \geq 3 $ ,
 +
a group structure can be uniquely introduced into the pointed sets $  \pi _{n} (X,\  A,\  x _{0} ) $ (
 +
and also into the sets $  \pi _{2} ( X ,\  x _{0} ) $ )
 +
so that all morphisms $  f _ \star  $
 +
are homomorphism (this structure accordingly corresponds to that described by formula (1)). On the other hand, the sets $  \pi _{2} ( X ,\  A ,\  x _{0} ) $
 +
if $  A \neq x _{0} $
 +
and $  \pi _{1} ( X ,\  x _{0} ) $
 +
carry only the inverse group operation. All this means that the above properties unambiguously define the homotopy groups (up to the order of multiplication in non-commutative groups).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
For any mapping $  u : \  ( I ^{n} ,\  \dot{I}  ^{n} ) \rightarrow ( X ,\  x _{2} ) $
 +
and any path $  \nu : \  I \rightarrow X $
 +
connecting two points $  x _{1} $
 +
and $  x _{2} $ ,
 +
the formula $  g _{t} (x) = \nu ( 1 - t ) ,\  x \in \dot{I}  ^{n} $ ,
 +
defines a homotopy of $  u \mid _ {\dot{I}  ^{n}} $ .
 +
By the homotopy extension axiom (cf. [[Cofibration|Cofibration]]) this homotopy can be extended to a homotopy $  u _{t} : \  I ^{n} \rightarrow X $
 +
for which $  u _{0} = u $ .
 +
The final mapping $  u _{1} $
 +
of this homotopy maps $  \dot{I}  ^{n} $
 +
into $  x _{1} $ ,
 +
i.e. represents a mapping $  ( I ^{n} ,\  \dot{I}  ^{n} ) \rightarrow ( X ,\  x _{1} ) $ .
 +
The corresponding element of the homotopy group depends only on the class $  [ u ] \in \pi _{n} ( X ,\  x _{2} ) $
 +
of $  u $
 +
and the homotopy class $  \alpha = [ \nu ] $
 +
of $  \nu $ ,
 +
and is denoted by the symbol $  \alpha x $ (
 +
if $  n = 1 $ ,
 +
by the symbol $  x ^ \alpha  $ ).
 +
The family $  G _{x} = \pi _{n} ( X ,\  x ) $
 +
is thus defined as a local family on the space $  X $ ,
 +
i.e. on the fundamental groupoid of this space. In particular, for any point $  x _{0} \in X $
 +
the group $  \pi _{1} ( X ,\  x _{0} ) $
 +
operates on $  \pi _{n} ( X ,\  x _{n} ) $ .  
 +
If $  n = 1 $
 +
these operators act as inner automorphisms: $  x ^ \alpha  = \alpha x \alpha ^{-1} $ ,
 +
and if $  n > 1 $
 +
they make the group $  \pi _{n} ( X ,\  x _{0} ) $
 +
into a $  \pi _{1} ( X ,\  x _{0} ) $ -
 +
module. For any continuous mapping $  f : \  ( X ,\  x _{0} ) \rightarrow ( Y ,\  y _{0} ) $
 +
the induced homomorphisms $  f _ \star  : \  \pi _{n} ( X ,\  x _{0} ) \rightarrow \pi _{n} ( Y ,\  y _{0} ) $
 +
are operator homomorphisms (homomorphisms of modules): $  f _ \star  ( \alpha x ) = f _ \star  ( \alpha ) f _ \star  (x) $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793033.png" /></td> </tr></table>
 
  
The resulting group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793034.png" /> is said to be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793035.png" />-th homotopy group (or the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793037.png" />-dimensional homotopy group) of the pointed pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793039.png" />; one also speaks of the homotopy group of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793040.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793041.png" /> or of the homotopy group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793042.png" /> with respect to the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793043.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793044.png" />. The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793045.png" /> are homomorphisms of these groups. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793046.png" /> it may be assumed that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793047.png" /> takes values in the category of groups (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793048.png" /> even in the category of Abelian groups).
+
In a similar way, the groups $  G _{x} = \pi _{n} ( X ,\  A ,\  x ) $ ,
 +
$  x \in A $ ,
 +
constitute a local family of homotopy groups on the subspace $  A $ .  
 +
In particular, the group $  \pi _{1} ( A ,\  x _{0} ) $
 +
operates on the homotopy group $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
so that if $  n > 2 $
 +
the group $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
is a $  \pi _{1} ( X ,\  A ,\  x _{0} ) $ -
 +
module. The group $  \pi _{2} ( X ,\  A ,\  x _{0} ) $
 +
is said to be a crossed $  ( \pi _{1} ( A ,\  x _{0} ) ,\  \partial ) $ -
 +
module (cf. [[Crossed modules|Crossed modules]]), where $  \partial : \  \pi _{2} ( X ,\  A ,\  x _{0} ) \rightarrow \pi _{1} ( A ,\  x _{0} ) $
 +
is the boundary homomorphism.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793049.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793050.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793051.png" />, or simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793052.png" />, and is called the absolute homotopy group of the pointed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793053.png" /> (or of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793054.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793055.png" />). Its elements are the homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793057.png" /> is the boundary of the cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793058.png" />. For such mappings formula (1) is meaningful for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793059.png" /> as well, and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793060.png" /> is a group. This group coincides with the classical fundamental group. The group operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793061.png" /> is usually called multiplication. This group is, generally speaking, non-Abelian, while the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793062.png" /> is Abelian. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793063.png" /> the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793064.png" /> and the corresponding homomorphisms form a functor from the category of pointed spaces into the category of groups (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793065.png" /> into the category of Abelian groups). This functor is the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793066.png" /> of the imbedding functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793067.png" /> and the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793068.png" /> described above.
+
The group $  \pi _{1} ( A ,\  x _{0} ) $
 +
acts as a group of operators not only on the groups $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
but also on the groups $  \pi _{n} ( A ,\  x _{0} ) $ ,  
 +
and also, by virtue of the natural homomorphism $  \pi _{1} ( A ,\  x _{0} ) \rightarrow \pi _{1} ( X ,\  x _{0} ) $ ,
 +
on the groups $  \pi _{n} ( X ,\  x _{0} ) $ .  
 +
With respect to these actions of $  \pi _{1} ( A ,\  x _{0} ) $
 +
all homomorphisms of the exact sequence $  \pi ( X ,\  A ,\  x _{0} ) $
 +
are operator homomorphisms, so that $  \pi _{1} ( A ,\  x _{0} ) $
 +
can be regarded as a group of operators on the sequence $  \pi ( X ,\  A ,\  x _{0} ) $ .  
 +
This is equivalent to saying that the sequences $  \pi ( X ,\  A ,\  x ) $ ,  
 +
$  x \in A $ ,  
 +
constitute a local family of exact sequences of the subspace $  A $ .
  
The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793069.png" /> is extended to include the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793071.png" /> is the pointed set of path-components (cf. [[Path-connected space|Path-connected space]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793072.png" />; the zero of this set is the component containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793073.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793074.png" /> is not defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793075.png" />. In order to simplify the formulations, the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793077.png" /> are usually also called homotopy groups, even though they are not groups in general.
 
  
For each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793078.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793079.png" /> represents a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793080.png" />, and thus defines a certain element of the homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793081.png" />. This element depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793082.png" /> and is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793083.png" />. The resulting mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793084.png" /> is a morphism of pointed sets (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793085.png" /> a homomorphism of groups) and is called a boundary homomorphism or a boundary operator. The boundary homomorphism, together with the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793087.png" /> induced by the imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793089.png" />, makes it possible to write down a sequence of groups and homomorphisms, infinite from the left:
+
If the complement $  X \setminus A $
 +
is represented as a union of disjoint open $  n $ -
 +
dimensional cells, then the $  \pi _{1} ( A ,\  x _{0} ) $ -
 +
module $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
is a free module (if $  n = 2 $ ,
 +
a free crossed module) and has a system of free generators — a basis in bijective (not necessarily natural) correspondence with the cells of $  X \setminus A $ (
 +
Whitehead's theorem).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793090.png" /></td> </tr></table>
+
The mappings $  ( I ^{n} ,\  \dot{I}  ^{n} ) \rightarrow ( X ,\  x _{0} ) $
 +
are in bijective correspondence with the mappings $  ( S ^{n} ,\  s _{0} ) \rightarrow ( X ,\  x _{0} ) $ ,
 +
where $  S ^{n} $
 +
is an $  n $ -
 +
dimensional sphere and $  s _{0} $
 +
is some point on it. For this reason the elements of $  \pi _{n} ( X ,\  x _{0} ) $
 +
can be regarded as the homotopy classes of mappings $  ( S ^{n} ,\  s _{0} ) \rightarrow ( X ,\  x _{0} ) $ .
 +
This is also true if $  n = 0 $ .
 +
The above identification depends on the selection of some relative homeomorphism $  \phi : \  ( I ^{n} ,\  \dot{I}  ^{n} ) \rightarrow ( S ^{n} ,\  s _{0} ) $ .
 +
It is common to select and fix the sphere $  S ^{n} $
 +
and the homeomorphism $  \phi $
 +
once and for all. In the original definition of Hurewicz, which is not frequently used nowadays, $  S ^{n} $
 +
was not fixed, while $  \phi $
 +
was given up to a homotopy. Such a specification of $  \phi $
 +
is equivalent to specifying an orientation on $  S ^{n} $ .
 +
Thus, according to Hurewicz, the elements of $  \pi _{n} ( X ,\  x _{0} ) $
 +
are pointed homotopy classes of mappings of an oriented $  n $ -
 +
dimensional sphere into $  X $ .
 +
The set $  [ S ^{n} ,\  X ] $
 +
of non-pointed homotopy classes of mappings $  S ^{n} \rightarrow X $
 +
is in bijective correspondence with the orbits of the action of $  \pi _{1} ( X ,\  x _{0} ) $
 +
on $  \pi _{n} ( X ,\  x _{0} ) $ (
 +
cf. [[Orbit|Orbit]]). If $  \pi _{1} ( X ,\  x _{0} ) = 0 $ (
 +
or, more generally, if $  \pi _{1} ( X ,\  x _{0} ) $
 +
acts trivially on $  \pi _{n} ( X ,\  x _{0} ) $ ),
 +
then $  X $
 +
is said to be homotopically $  n $ -
 +
simple. In this case $  \pi _{n} ( X ,\  x _{0} ) $
 +
is independent of $  x _{0} $ (
 +
so that the notation $  \pi _{n} ( X ) $
 +
is fully justified). This group is naturally identified with the set $  [ S ^{n} ,\  X ] $ ,
 +
which, as a consequence, has a group structure. A space that is homotopically $  n $ -
 +
simple for all $  n $
 +
is said to be Abelian.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793091.png" /></td> </tr></table>
+
Let $  s _{n} $
 +
be the orientation class of the sphere $  S ^{n} $
 +
and let $  h ( [ f ] ) = f _ \star  ( s _{n} ) $ ,
 +
$  [ f ] \in \pi _{n} ( x ,\  x _{0} ) $ .
 +
This defines a homomorphism $  h : \  \pi _{n} ( X ,\  x _{0} ) \rightarrow H _{n} ( X ) $ ,
 +
the so-called Hurewicz homomorphism. Its kernel contains all elements of the form $  \alpha x - x $ ,
 +
$  x \in \pi _{n} ( X ,\  x _{0} ) $ ,
 +
$  \alpha \in \pi _{1} ( X ,\  x _{0} ) $ (
 +
if $  n = 1 $ ,
 +
all elements of the form $  x ^ \alpha  x ^{-1} = \alpha x \alpha ^{-1} x ^{-1} $ ,
 +
i.e. it contains the commutator $  [ \pi _{1} ,\  \pi _{1} ] $
 +
of $  \pi _{1} ( X ,\  x _{0} ) $ ).
 +
Poincaré's classical theorem states that for $  n = 1 $
 +
the kernel of $  h $
 +
coincides with the commutator $  [ \pi _{1} ,\  \pi _{1} ] $ ,
 +
so that the group $  H _{1} (X) $
 +
is isomorphic to the Abelianization of the fundamental group $  \pi _{1} ( X ,\  x _{0} ) $ .
 +
Hurewicz's theorem, which is a generalization of Poincaré's theorem to the case $  n > 1 $ ,
 +
states that if $  \pi _{i} (X) = 0 $
 +
for $  i < n $ ,
 +
then the homomorphism $  h : \  \pi _{n} (X) \rightarrow H _{n} (X) $
 +
is an isomorphism (and the homomorphism $  h : \  \pi _{n+1} (X) \rightarrow H _{n+1} (X) $
 +
is an epimorphism).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793092.png" /></td> </tr></table>
+
In a similar way, the elements of $  \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
can be regarded as (pointed) homotopy classes of mappings $  ( E ,\  S ) \rightarrow ( X ,\  A ) $ ,
 +
where $  E $
 +
is an (oriented) $  n $ -
 +
dimensional ball and $  S $
 +
is its boundary. If the pair $  ( X ,\  A ) $
 +
is homotopically $  n $ -
 +
simple (i.e. if $  \pi _{1} ( A ,\  x _{0} ) $
 +
acts trivially on $  \pi _{n} ( X ,\  A ,\  x _{0} ) $ ),
 +
then the requirement of pointedness may be dropped in this definition. The formula$$
 +
h ( [ f ] )  =  f _ \star  ( e _{n} ) ,
 +
$$
 +
where $  e _{n} $
 +
is the orientation class of the pair $  ( E ,\  S ) $
 +
and $  [ f ] \in \pi _{n} ( X ,\  A ,\  x _{0} ) $
 +
defines the Hurewicz homomorphism$$
 +
h : \  \pi _{n} ( X ,\  A ,\  x _{0} )  \rightarrow 
 +
H _{n} ( X ,\  A ) .
 +
$$
 +
If $  \pi _{1} ( A ,\  x _{0} ) = 0 $
 +
and $  \pi _{n} ( X ,\  A ,\  x _{0} ) = 0 $
 +
for $  i < n $ ,
 +
this homomorphism is an isomorphism (Hurewicz's theorem for relative groups).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793093.png" /></td> </tr></table>
+
Two principal methods are known for the computation of the homotopy groups of specific spaces: the method of killing spaces (cf. [[Killing space|Killing space]]) and the method of homotopy resolutions (cf. [[Homotopy type|Homotopy type]]; [[Postnikov system|Postnikov system]]). The first method is based on the isomorphism $  \pi _{n+1} (X) \approx H _{n+1} ( X ,\  n ) $ ,
 +
which follows from Hurewicz's theorem and the definition of the killing space $  ( X ,\  n ) $ .
 +
This isomorphism reduces the computation of $  \pi _{n+1} (X) $
 +
to the problem of computing the homology groups $  H _{n+1} ( X ,\  n ) $ .
 +
The space $  ( X ,\  n ) $
 +
fibres over the space $  ( X ,\  n - 1 ) $
 +
with fibre $  K ( \pi _{n} (X) ,\  n - 1 ) $ ,
 +
and the homology groups of the space $  K ( \pi ,\  n ) $
 +
are known. Therefore one may try to find the lower homology groups of killing spaces by induction. The problem of computing the homology groups of a fibre space from the homology groups of its base and fibre is still not completely solved in its general formulation (and, obviously, a general satisfactory solution does not exist). However, extensive information on the homology groups of the spaces $  ( X ,\  n ) $
 +
can be extracted from the corresponding Serre spectral sequence. In many cases this information is sufficient for the computation of $  H _{n+1} ( X ,\  n ) \approx \pi _{n+1} (X) $ ,
 +
at least for some $  n $ .
 +
An essential technical simplification of the problem is obtained on the basis of the Serre's theory of classes of Abelian groups and the $  G _{p} $ -
 +
approximation derived from it. With this theory it is possible to compute entirely in the cohomology and only for the coefficient groups $  \mathbf Z / p $ .
 +
The geometric principles on which this technique is based were first clarified by J.F. Adams and D. Sullivan on the basis of the concept of localization of topological spaces at a given prime number $  p $ .
  
This is an [[Exact sequence|exact sequence]]; it is called the exact homotopy sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793094.png" /> and is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793095.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793096.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793097.png" />, then the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793098.png" /> is an isomorphism (also for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h04793099.png" />).
 
  
The boundary homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930100.png" /> is natural, that is, it is a morphism of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930101.png" /> into the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930102.png" /> (more exactly, into the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930103.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930104.png" />). This makes it possible to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930105.png" /> as a functor that takes values in the category of exact sequences of pointed sets which, except for the last six sets, are Abelian groups and, except for the last three sets, are groups.
+
The second (also inductive) method of computing homotopy groups consists of a stepwise construction of the homotopy resolution of the space $  X $ .  
 +
Suppose the $  n $ -
 +
th term of this resolution is known (e.g. if $  X = S ^{n} $ ,
 +
then $  X _{n} = K ( \mathbf Z ,\  n ) $ ).  
 +
The next term must be the fibre space over $  X _{n} $
 +
with fibre $  K ( \pi _{n+1} (X) ,\  n + 1 ) $ ;
 +
moreover, the group $  H _{n+1} $
 +
must be isomorphic to the known group $  H _{n+1} (X) $ .  
 +
This gives (on the basis of the corresponding spectral sequence) definite information on the group $  \pi _{n+1} (X) $ ,
 +
which, in many cases, makes it possible to compute it completely. For example, for $  X = S ^{n} $
 +
by this method all groups $  \pi _{n+k} ( S ^{n} ) $ ,
 +
$  k \leq 13 $ ,  
 +
can be found. In its modern form, this method is also based on the concept of localization.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930106.png" /> be an arbitrary [[Fibration|fibration]] in the sense of Serre and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930109.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930110.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930111.png" /> defines a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930112.png" /> of pointed pairs. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930113.png" /> the induced homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930114.png" /> is an isomorphism. In particular, this is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930115.png" />. In the latter case the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930116.png" /> unambiguously defines a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930117.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930118.png" /> is the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930119.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930120.png" />. This homomorphisms is called the homotopy transgression. It occurs in the exact sequence
+
The method of homology resolutions was extended (cf. [[#References|[4]]]) to an algorithm that is applicable to any simply-connected finite $  \mathop{\rm CW}\nolimits $ -
 
+
complex and that gives all its homotopy groups. However, for practical use this algorithm is too complicated.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930121.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930122.png" /></td> </tr></table>
 
 
 
This sequence is called the homotopy sequence of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930123.png" />. Putting a fibration into correspondence with its homotopy sequence yields a functor on the category of all (pointed) fibrations.
 
 
 
In the particular case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930124.png" /> is the standard [[Serre fibration|Serre fibration]] of paths over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930125.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930126.png" /> one has the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930127.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930128.png" /> is the [[Loop space|loop space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930129.png" />. This isomorphism is called the Hurewicz isomorphism.
 
 
 
The above properties actually unambiguously define the homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930130.png" />, i.e. may be taken as axioms which describe these groups. In fact, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930131.png" /> be an arbitrary sequence of homotopy-invariant normalized functors, defined on the category of pointed spaces, taking values in the category of pointed sets, and having the following property: For any fibration in the sense of Serre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930132.png" />, any subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930133.png" /> and any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930134.png" />, the induced homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930135.png" /> is an isomorphism. Such a sequence is called a homotopy system if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930136.png" /> there is defined a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930137.png" /> of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930138.png" /> into the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930139.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930140.png" />, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930141.png" />) that is an isomorphism for any pointed pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930142.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930143.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930144.png" />. Any homotopy system is isomorphic to the homotopy system constructed above, which consists of homotopy groups. Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930145.png" />, a group structure can be uniquely introduced into the pointed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930146.png" /> (and also into the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930147.png" />) so that all morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930148.png" /> are homomorphism (this structure accordingly corresponds to that described by formula (1)). On the other hand, the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930149.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930151.png" /> carry only the inverse group operation. All this means that the above properties unambiguously define the homotopy groups (up to the order of multiplication in non-commutative groups).
 
 
 
For any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930152.png" /> and any path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930153.png" /> connecting two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930155.png" />, the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930156.png" />, defines a homotopy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930157.png" />. By the homotopy extension axiom (cf. [[Cofibration|Cofibration]]) this homotopy can be extended to a homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930158.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930159.png" />. The final mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930160.png" /> of this homotopy maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930161.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930162.png" />, i.e. represents a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930163.png" />. The corresponding element of the homotopy group depends only on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930164.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930165.png" /> and the homotopy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930166.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930167.png" />, and is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930168.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930169.png" />, by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930170.png" />). The family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930171.png" /> is thus defined as a local family on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930172.png" />, i.e. on the fundamental groupoid of this space. In particular, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930173.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930174.png" /> operates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930175.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930176.png" /> these operators act as inner automorphisms: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930177.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930178.png" /> they make the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930179.png" /> into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930180.png" />-module. For any continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930181.png" /> the induced homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930182.png" /> are operator homomorphisms (homomorphisms of modules): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930183.png" />.
 
 
 
In a similar way, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930185.png" />, constitute a local family of homotopy groups on the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930186.png" />. In particular, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930187.png" /> operates on the homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930188.png" /> so that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930189.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930190.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930191.png" />-module. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930192.png" /> is said to be a crossed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930194.png" />-module (cf. [[Crossed modules|Crossed modules]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930195.png" /> is the boundary homomorphism.
 
 
 
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930196.png" /> acts as a group of operators not only on the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930197.png" /> but also on the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930198.png" />, and also, by virtue of the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930199.png" />, on the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930200.png" />. With respect to these actions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930201.png" /> all homomorphisms of the exact sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930202.png" /> are operator homomorphisms, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930203.png" /> can be regarded as a group of operators on the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930204.png" />. This is equivalent to saying that the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930206.png" />, constitute a local family of exact sequences of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930207.png" />.
 
 
 
If the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930208.png" /> is represented as a union of disjoint open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930209.png" />-dimensional cells, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930210.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930211.png" /> is a free module (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930212.png" />, a free crossed module) and has a system of free generators — a basis in bijective (not necessarily natural) correspondence with the cells of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930213.png" /> (Whitehead's theorem).
 
 
 
The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930214.png" /> are in bijective correspondence with the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930215.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930216.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930217.png" />-dimensional sphere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930218.png" /> is some point on it. For this reason the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930219.png" /> can be regarded as the homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930220.png" />. This is also true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930221.png" />. The above identification depends on the selection of some relative homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930222.png" />. It is common to select and fix the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930223.png" /> and the homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930224.png" /> once and for all. In the original definition of Hurewicz, which is not frequently used nowadays, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930225.png" /> was not fixed, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930226.png" /> was given up to a homotopy. Such a specification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930227.png" /> is equivalent to specifying an orientation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930228.png" />. Thus, according to Hurewicz, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930229.png" /> are pointed homotopy classes of mappings of an oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930230.png" />-dimensional sphere into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930231.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930232.png" /> of non-pointed homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930233.png" /> is in bijective correspondence with the orbits of the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930234.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930235.png" /> (cf. [[Orbit|Orbit]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930236.png" /> (or, more generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930237.png" /> acts trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930238.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930239.png" /> is said to be homotopically <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930241.png" />-simple. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930242.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930243.png" /> (so that the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930244.png" /> is fully justified). This group is naturally identified with the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930245.png" />, which, as a consequence, has a group structure. A space that is homotopically <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930246.png" />-simple for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930247.png" /> is said to be Abelian.
 
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930248.png" /> be the orientation class of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930249.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930250.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930251.png" />. This defines a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930252.png" />, the so-called Hurewicz homomorphism. Its kernel contains all elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930253.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930254.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930255.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930256.png" />, all elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930257.png" />, i.e. it contains the commutator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930258.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930259.png" />). Poincaré's classical theorem states that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930260.png" /> the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930261.png" /> coincides with the commutator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930262.png" />, so that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930263.png" /> is isomorphic to the Abelianization of the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930264.png" />. Hurewicz's theorem, which is a generalization of Poincaré's theorem to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930265.png" />, states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930266.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930267.png" />, then the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930268.png" /> is an isomorphism (and the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930269.png" /> is an epimorphism).
 
 
 
In a similar way, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930270.png" /> can be regarded as (pointed) homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930271.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930272.png" /> is an (oriented) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930273.png" />-dimensional ball and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930274.png" /> is its boundary. If the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930275.png" /> is homotopically <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930276.png" />-simple (i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930277.png" /> acts trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930278.png" />), then the requirement of pointedness may be dropped in this definition. The formula
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930279.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930280.png" /> is the orientation class of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930281.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930282.png" /> defines the Hurewicz homomorphism
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930283.png" /></td> </tr></table>
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930284.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930285.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930286.png" />, this homomorphism is an isomorphism (Hurewicz's theorem for relative groups).
 
 
 
Two principal methods are known for the computation of the homotopy groups of specific spaces: the method of killing spaces (cf. [[Killing space|Killing space]]) and the method of homotopy resolutions (cf. [[Homotopy type|Homotopy type]]; [[Postnikov system|Postnikov system]]). The first method is based on the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930287.png" />, which follows from Hurewicz's theorem and the definition of the killing space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930288.png" />. This isomorphism reduces the computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930289.png" /> to the problem of computing the homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930290.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930291.png" /> fibres over the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930292.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930293.png" />, and the homology groups of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930294.png" /> are known. Therefore one may try to find the lower homology groups of killing spaces by induction. The problem of computing the homology groups of a fibre space from the homology groups of its base and fibre is still not completely solved in its general formulation (and, obviously, a general satisfactory solution does not exist). However, extensive information on the homology groups of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930295.png" /> can be extracted from the corresponding Serre spectral sequence. In many cases this information is sufficient for the computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930296.png" />, at least for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930297.png" />. An essential technical simplification of the problem is obtained on the basis of the Serre's theory of classes of Abelian groups and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930298.png" />-approximation derived from it. With this theory it is possible to compute entirely in the cohomology and only for the coefficient groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930299.png" />. The geometric principles on which this technique is based were first clarified by J.F. Adams and D. Sullivan on the basis of the concept of localization of topological spaces at a given prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930300.png" />.
 
 
 
The second (also inductive) method of computing homotopy groups consists of a stepwise construction of the homotopy resolution of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930301.png" />. Suppose the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930302.png" />-th term of this resolution is known (e.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930303.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930304.png" />). The next term must be the fibre space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930305.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930306.png" />; moreover, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930307.png" /> must be isomorphic to the known group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930308.png" />. This gives (on the basis of the corresponding spectral sequence) definite information on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930309.png" />, which, in many cases, makes it possible to compute it completely. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930310.png" /> by this method all groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930311.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930312.png" />, can be found. In its modern form, this method is also based on the concept of localization.
 
 
 
The method of homology resolutions was extended (cf. [[#References|[4]]]) to an algorithm that is applicable to any simply-connected finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930313.png" />-complex and that gives all its homotopy groups. However, for practical use this algorithm is too complicated.
 
  
 
Since the homotopy theory is completely equivalent to the homotopy theory of simplicial sets, the definition of a homotopy group may be transferred to any (complete) simplicial set. The  "combinatorial"  definition obtained (due to D. Kan) can easily be extended to an algorithm. However, this algorithm is also too complicated for practical use.
 
Since the homotopy theory is completely equivalent to the homotopy theory of simplicial sets, the definition of a homotopy group may be transferred to any (complete) simplicial set. The  "combinatorial"  definition obtained (due to D. Kan) can easily be extended to an algorithm. However, this algorithm is also too complicated for practical use.
  
From any of the above methods it is easy to establish that the homotopy groups of a simply-connected space having finitely-generated homology groups, are also finitely generated. The analogous statement for non-simply connected spaces (i.e. its homology groups should be finitely generated as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930314.png" />-modules) is, in general, not true.
+
From any of the above methods it is easy to establish that the homotopy groups of a simply-connected space having finitely-generated homology groups, are also finitely generated. The analogous statement for non-simply connected spaces (i.e. its homology groups should be finitely generated as $  \pi _{1} (X) $ -
 
+
modules) is, in general, not true.
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930315.png" /> be the (reduced) [[Suspension|suspension]] functor, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930316.png" /> be the loop functor. Since these functors are adjoint, the identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930317.png" /> defines an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930318.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930319.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930320.png" />, this imbedding defines a homomorphism
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930321.png" /></td> </tr></table>
 
 
 
which is known as the suspension homomorphism. It coincides with the homomorphism obtained by assigning to an arbitrary (pointed) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930322.png" /> its suspension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930323.png" />. This homomorphism occurs in an exact sequence:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930324.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930325.png" /></td> </tr></table>
 
  
This sequence is called the suspension sequence of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930326.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930327.png" /> in it is a generalization of the classical [[Hopf invariant|Hopf invariant]].
+
Let $  S $
 +
be the (reduced) [[Suspension|suspension]] functor, and let $  \Omega $
 +
be the loop functor. Since these functors are adjoint, the identity mapping $  S X \rightarrow S X $
 +
defines an imbedding $  X \subset \Omega S X $ ,
 +
for any $  X $ .
 +
Since $  \pi _{n} ( \Omega S X ) \approx \pi _{n+1} ( S X ) $ ,
 +
this imbedding defines a homomorphism$$
 +
E : \  \pi _{n} ( X )  \rightarrow  \pi _{n+1} ( S X ) ,
 +
$$
 +
which is known as the suspension homomorphism. It coincides with the homomorphism obtained by assigning to an arbitrary (pointed) mapping $  f : \  S ^{n} \rightarrow X $
 +
its suspension $  S f : \  S ^{n+1} \rightarrow S X $ .
 +
This homomorphism occurs in an exact sequence:$$
 +
{} \dots \rightarrow  \pi _{n} (X)  \rightarrow ^ E 
 +
\pi _{n+1} ( S X )  \rightarrow ^ H  \pi _{n}
 +
( \Omega S X ,\  X )    \stackrel \partial  \rightarrow 
 +
$$
 +
$$
 +
\stackrel \partial  \rightarrow    \pi _{n-1} (X)  \rightarrow \dots .
 +
$$
 +
This sequence is called the suspension sequence of the space $  X $ .  
 +
The homomorphism $  H $
 +
in it is a generalization of the classical [[Hopf invariant|Hopf invariant]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930328.png" /> is a countable CW-complex with one vertex, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930329.png" /> may be replaced by the infinite reduced product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930330.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930331.png" />. This shows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930332.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930333.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930334.png" /> is an isomorphism for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930335.png" /> and an epimorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930336.png" />. This theorem is known as Freudenthal's suspension theorem (H. Freudenthal first published the proof for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930337.png" />, although the theorem was known much earlier.)
+
If $  X $
 +
is a countable CW-complex with one vertex, the space $  \Omega SX $
 +
may be replaced by the infinite reduced product $  X _ \infty  $
 +
of the complex $  X $ .  
 +
This shows that if $  \pi _{i} (X) = 0 $
 +
for $  i \leq m $ ,  
 +
then $  E $
 +
is an isomorphism for all $  n \leq 2m - 1 $
 +
and an epimorphism if $  n = 2m - 1 $ .  
 +
This theorem is known as Freudenthal's suspension theorem (H. Freudenthal first published the proof for the case $  X = S ^{n} $ ,  
 +
although the theorem was known much earlier.)
  
Freudenthal's theorem shows that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930338.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930339.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930340.png" />. It is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930342.png" />-th stable homotopy group of the sphere (cf. also [[Stable homotopy group|Stable homotopy group]]). Similar stabilization phenomena occur for the homotopy groups of the orthogonal groups, of the Thom spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930343.png" /> (cf. [[Thom space|Thom space]]) and in many other cases. The general study of these phenomena is most conveniently done within the framework of the so-called theory of spectra. In this theory stable homotopy groups arise as the homotopy groups of spectra. These groups have an essentially simpler structure than the homotopy groups of a space and their study (and computation) is an easier task. For example, for the computation of these groups one has a special device: the Adams [[Spectral sequence|spectral sequence]].
+
Freudenthal's theorem shows that for $  k \leq 2n - 1 $
 +
the group $  \pi _{n+k} (S ^{n} ) $
 +
is independent of $  n $ .  
 +
It is called the $  k $ -
 +
th stable homotopy group of the sphere (cf. also [[Stable homotopy group|Stable homotopy group]]). Similar stabilization phenomena occur for the homotopy groups of the orthogonal groups, of the Thom spaces $  \mathop{\rm MSO}\nolimits (n) $ (
 +
cf. [[Thom space|Thom space]]) and in many other cases. The general study of these phenomena is most conveniently done within the framework of the so-called theory of spectra. In this theory stable homotopy groups arise as the homotopy groups of spectra. These groups have an essentially simpler structure than the homotopy groups of a space and their study (and computation) is an easier task. For example, for the computation of these groups one has a special device: the Adams [[Spectral sequence|spectral sequence]].
  
Homotopy groups have been generalized in various directions. For example, an attempt was made to replace the spheres by other spaces. Here one may note toroidal homotopy groups, obtained by interpreting the [[Whitehead product|Whitehead product]] as a commutator. It was also shown that the set of homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930344.png" /> admits a group operation which is natural with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930345.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930346.png" /> is a [[Co-H-space|co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930347.png" />-space]]. Homotopy groups with coefficients were obtained by replacing the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930348.png" /> by the Moore spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930349.png" /> (cf. [[Moore space|Moore space]]). This definition of homotopy groups with coefficients was not very successful. A more satisfactory definition (compatible with the general Eckmann–Hilton duality principle) was obtained by replacing the Moore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930350.png" />-spaces by co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930351.png" />-spaces. However, these homotopy groups were not defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930352.png" /> (e.g. for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930353.png" /> the additive group of real numbers, these groups are not defined).
+
Homotopy groups have been generalized in various directions. For example, an attempt was made to replace the spheres by other spaces. Here one may note toroidal homotopy groups, obtained by interpreting the [[Whitehead product|Whitehead product]] as a commutator. It was also shown that the set of homotopy classes of mappings $  X \rightarrow Y $
 +
admits a group operation which is natural with respect to $  Y $
 +
if and only if $  X $
 +
is a [[Co-H-space|co-$  H $ -
 +
space]]. Homotopy groups with coefficients were obtained by replacing the spheres $  S ^{n} $
 +
by the Moore spaces $  M (G,\  n) $ (
 +
cf. [[Moore space|Moore space]]). This definition of homotopy groups with coefficients was not very successful. A more satisfactory definition (compatible with the general Eckmann–Hilton duality principle) was obtained by replacing the Moore $  M $ -
 +
spaces by co-$  M $ -
 +
spaces. However, these homotopy groups were not defined for all $  G $ (
 +
e.g. for $  G $
 +
the additive group of real numbers, these groups are not defined).
  
The question of the construction of homotopy groups in categories other than the category of pointed pairs has been studied in detail. First of all one has to mention the homotopy groups of a triad (cf. [[Triads|Triads]], see, e.g., [[#References|[3]]]), which were very useful in the study of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930354.png" />. A very general construction of homotopy groups was proposed in connection with studies on duality. On the basis of the concept of a standard construction (see [[#References|[6]]]) the construction of homotopy groups was transferred to arbitrary categories. A fundamental role in this construction is played by the homotopy groups of simplicial sets mentioned earlier.
+
The question of the construction of homotopy groups in categories other than the category of pointed pairs has been studied in detail. First of all one has to mention the homotopy groups of a triad (cf. [[Triads|Triads]], see, e.g., [[#References|[3]]]), which were very useful in the study of the homomorphism $  E $ .  
 +
A very general construction of homotopy groups was proposed in connection with studies on duality. On the basis of the concept of a standard construction (see [[#References|[6]]]) the construction of homotopy groups was transferred to arbitrary categories. A fundamental role in this construction is played by the homotopy groups of simplicial sets mentioned earlier.
  
 
====References====
 
====References====
Line 123: Line 469:
 
was the first to study the higher homotopy groups in detail, the definition was in fact suggested a few years earlier by E. Čech [[#References|[a2]]]. The action of the fundamental group on the higher homotopy groups was first studied by S. Eilenberg [[#References|[a3]]]. A good general reference for homotopy groups is [[#References|[a4]]].
 
was the first to study the higher homotopy groups in detail, the definition was in fact suggested a few years earlier by E. Čech [[#References|[a2]]]. The action of the fundamental group on the higher homotopy groups was first studied by S. Eilenberg [[#References|[a3]]]. A good general reference for homotopy groups is [[#References|[a4]]].
  
The stable homotopy groups form a generalized homology theory, i.e. a theory which satisfies all the Eilenberg–Steenrod axioms except possibly the dimension axiom. This theory is in fact defined by the spectrum of spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047930/h047930355.png" />, cf. [[Spectrum of spaces|Spectrum of spaces]]. The corresponding generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) defined by this spectrum consists of the cohomotopy group. Cf., e.g., [[#References|[a4]]] and [[#References|[a11]]] for more details. Powerful tools for computing the stable homotopy groups of the spheres (besides the (classical) Adams spectral sequence) involve the Adams–Novikov spectral sequence, the so-called chromatic spectral sequence and complex [[Cobordism|cobordism]], cf. [[#References|[a12]]].
+
The stable homotopy groups form a generalized homology theory, i.e. a theory which satisfies all the Eilenberg–Steenrod axioms except possibly the dimension axiom. This theory is in fact defined by the spectrum of spheres $  ( S ^{n} ) _{n} $ ,  
 +
cf. [[Spectrum of spaces|Spectrum of spaces]]. The corresponding generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) defined by this spectrum consists of the cohomotopy group. Cf., e.g., [[#References|[a4]]] and [[#References|[a11]]] for more details. Powerful tools for computing the stable homotopy groups of the spheres (besides the (classical) Adams spectral sequence) involve the Adams–Novikov spectral sequence, the so-called chromatic spectral sequence and complex [[Cobordism|cobordism]], cf. [[#References|[a12]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1a]</TD> <TD valign="top">  W. Hurewicz,  "Beiträge zur Topologie der Deformationen I-II"  ''Proc. Ned. Akad. Weten. Ser. A'' , '''38'''  (1935)  pp. 112–119; 521–528</TD></TR><TR><TD valign="top">[a1b]</TD> <TD valign="top">  W. Hurewicz,  "Beiträge zur Topologie der Deformationen III-IV"  ''Proc. Ned. Akad. Weten. Ser. A'' , '''39'''  (1936)  pp. 117–126; 215–224</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Čech,  "Höherdimensionale Homotopiegruppen" , ''Verh. Intern. Mathematikerkongress Zürich, 1932'' , O. Füssli  (1932)  pp. 203</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Eilenberg,  "On the relation between the fundamental group of a space and the higher homotopy groups"  ''Fund. Math.'' , '''32'''  (1939)  pp. 167–175</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)  pp. 23; 415–455</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Gray,  "Homotopy theory. An introduction to algebraic topology" , Acad. Press  (1975)  pp. §12</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.J. Hilton,  "An introduction to homotopy theory" , Cambridge Univ. Press  (1953)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1960)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D. Sullivan,  "Genetics of homotopy theory and the Adams conjecture"  ''Ann. of Math.'' , '''100'''  (1974)  pp. 1–79</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D.G. Quillen,  "Homotopical algebra" , Springer  (1967)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  B. Eckmann,  "Homotopie et dualité" , ''Coll. Topol. Algébrique Louvain, 1956'' , Masson  (1957)  pp. 41–53</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D.C. Ravenel,  "Complex cobordism and stable homotopy groups of spheres" , Acad. Press  (1986)</TD></TR></table>
 
<table><TR><TD valign="top">[a1a]</TD> <TD valign="top">  W. Hurewicz,  "Beiträge zur Topologie der Deformationen I-II"  ''Proc. Ned. Akad. Weten. Ser. A'' , '''38'''  (1935)  pp. 112–119; 521–528</TD></TR><TR><TD valign="top">[a1b]</TD> <TD valign="top">  W. Hurewicz,  "Beiträge zur Topologie der Deformationen III-IV"  ''Proc. Ned. Akad. Weten. Ser. A'' , '''39'''  (1936)  pp. 117–126; 215–224</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Čech,  "Höherdimensionale Homotopiegruppen" , ''Verh. Intern. Mathematikerkongress Zürich, 1932'' , O. Füssli  (1932)  pp. 203</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Eilenberg,  "On the relation between the fundamental group of a space and the higher homotopy groups"  ''Fund. Math.'' , '''32'''  (1939)  pp. 167–175</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)  pp. 23; 415–455</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Gray,  "Homotopy theory. An introduction to algebraic topology" , Acad. Press  (1975)  pp. §12</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.J. Hilton,  "An introduction to homotopy theory" , Cambridge Univ. Press  (1953)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1960)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D. Sullivan,  "Genetics of homotopy theory and the Adams conjecture"  ''Ann. of Math.'' , '''100'''  (1974)  pp. 1–79</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D.G. Quillen,  "Homotopical algebra" , Springer  (1967)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  B. Eckmann,  "Homotopie et dualité" , ''Coll. Topol. Algébrique Louvain, 1956'' , Masson  (1957)  pp. 41–53</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D.C. Ravenel,  "Complex cobordism and stable homotopy groups of spheres" , Acad. Press  (1986)</TD></TR></table>

Latest revision as of 10:04, 15 December 2019

A generalization of the fundamental group, proposed by W. Hurewicz [1] in the context of problems on the classification of continuous mappings. Homotopy groups are defined for any $ n \geq 1 $ . For $ n = 1 $ the homotopy group is identical with the fundamental group. The definition of homotopy groups is not constructive and for this reason their computation is a difficult task, general methods for which were developed only in the 1950s. Their importance is due to the fact that all problems in homotopy theory can be reduced (cf. Homotopy type), to a greater or lesser extent, to the computation of certain homotopy groups.

Let$$ I ^{n} = \{ {( t _{1} \dots t _{n} )} : { 0 \leq t _{1} \leq 1 \dots 0 \leq t _{n} \leq 1} \} $$ be the dimensional unit cube, let $ I _{n-1} $ be its face $ t _{n} = 0 $ , and let $ J ^{n-1} $ be the union of its remaining faces. For any pointed pair $ ( X ,\ A ,\ x _{0} ) $ ( cf. Pointed object) the symbol $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ ( or simply $ \pi _{n} ( X ,\ A ) $ ) denotes the pointed set of all homotopy classes (cf. Homotopy) of mappings$$ u : \ ( I ^{n} ,\ I ^{n-1} ,\ J ^{n-1} ) \rightarrow ( X ,\ A ,\ x _{0} ) . $$ The distinguished (zero) element of this set is the constant mapping that maps the whole cube $ I ^{n} $ into $ x _{0} $ . Any continuous mapping$$ f : \ ( X ,\ A ,\ x _{0} ) \rightarrow ( Y ,\ B ,\ y _{0} ) $$ induces a morphism$$ f _ \star : \ \pi _{n} ( X ,\ A ,\ x _{0} ) \rightarrow \pi _{n} ( Y ,\ B ,\ y _{0} ) $$ of pointed sets. For any $ n \geq 1 $ the sets $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ and the morphisms $ f _ \star $ constitute a functor $ \pi _{n} $ from the category of pointed pairs into the category of pointed sets. This functor is homotopy invariant, i.e. $ f _ \star = g _ \star $ if $ f $ and $ g $ are homotopic (as mappings of pointed pairs). Furthermore, it is normalized in the sense that if $ X = A = x _{0} $ , then $ \pi _{n} ( X ,\ A ,\ x _{0} ) = 0 $ .


For $ n \geq 2 $ it is possible to introduce into the set $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ an operation of addition, with respect to which it becomes a group (if $ n \geq 2 $ even an Abelian group). By definition, if $ x = [ u ] $ and $ y = [ v ] $ , then $ x + y = [ w ] $ , where $ w $ is the mapping$$ ( I ^{n} ,\ I ^{n-1} ,\ J ^{n-1} ) \rightarrow ( X ,\ A ,\ x _{0} ) $$ defined by the formula$$ \tag{1} w ( t _{1} \dots t _{n} ) = $$ $$ = \left \{ \begin{array}{ll} u ( 2 t _{1} ,\ t _{2} \dots t _{n} ) & \textrm{ if } 0 \leq t _{1} \leq 1 / 2 , \\ v ( 2 t _{1} -1 ,\ t _{2} \dots t _{n} ) & \textrm{ if } 1 / 2 \leq t _{1} \leq 1 . \\ \end{array} \right .$$ The resulting group $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ is said to be the $ n $ - th homotopy group (or the $ n $ - dimensional homotopy group) of the pointed pair $ ( X ,\ A ,\ x _{0} ) $ ; one also speaks of the homotopy group of the pair $ ( X ,\ A ) $ at $ x _{0} $ or of the homotopy group of the space $ X $ with respect to the subspace $ A $ at $ x _{0} $ . The mappings $ f _ \star $ are homomorphisms of these groups. Thus, if $ n \geq 2 $ it may be assumed that the function $ \pi _{n} $ takes values in the category of groups (if $ n > 2 $ even in the category of Abelian groups).

For $ A = x _{0} $ the group $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ is denoted by $ \pi _{n} ( X ,\ x _{0} ) $ , or simply by $ \pi _{n} (X) $ , and is called the absolute homotopy group of the pointed space $ ( X ,\ x _{0} ) $ ( or of the space $ X $ at $ x _{0} $ ). Its elements are the homotopy classes of mappings $ ( I ^{n} ,\ \dot{I} ^{n} ) \rightarrow ( X ,\ x _{0} ) $ , where $ \dot{I} ^{n} = I ^{n-1} \cup J ^{n-1} $ is the boundary of the cube $ I ^{n} $ . For such mappings formula (1) is meaningful for $ n = 1 $ as well, and so $ \pi _{n} ( X ,\ x _{0} ) $ is a group. This group coincides with the classical fundamental group. The group operation in $ \pi _{n} ( X ,\ x _{0} ) $ is usually called multiplication. This group is, generally speaking, non-Abelian, while the group $ \pi _{2} ( X ,\ x _{0} ) $ is Abelian. For any $ n \geq 1 $ the groups $ \pi _{n} ( X ,\ x _{0} ) $ and the corresponding homomorphisms form a functor from the category of pointed spaces into the category of groups (if $ n > 1 $ into the category of Abelian groups). This functor is the composition $ \pi _{n} \circ \iota $ of the imbedding functor $ \iota : \ ( X ,\ x _{0} ) \rightarrow ( X ,\ x _{0} ,\ x _{0} ) $ and the functor $ \pi _{n} $ described above.

The functor $ \pi _{n} \circ \iota $ is extended to include the case $ n = 0 $ , where $ \pi _{0} ( X ,\ x _{0} ) $ is the pointed set of path-components (cf. Path-connected space) of $ X $ ; the zero of this set is the component containing $ x _{0} $ . The set $ \pi _{0} ( X ,\ A ,\ x _{0} ) $ is not defined for $ A \neq x _{0} $ . In order to simplify the formulations, the sets $ \pi _{0} ( X ,\ x _{0} ) $ and $ \pi _{1} ( X ,\ A ,\ x _{0} ) $ are usually also called homotopy groups, even though they are not groups in general.

For each element $ x = [ u ] \in \pi _{n} ( X ,\ A ,\ x _{0} ) $ the mapping $ u \mid _ {I ^{n-1}} $ represents a mapping $ ( I ^{n-1} ,\ \dot{I} ^{n-1} ) \rightarrow ( A ,\ x _{0} ) $ , and thus defines a certain element of the homotopy group $ \pi _{n-1} ( A ,\ x _{0} ) $ . This element depends only on $ x $ and is denoted by the symbol $ \partial x $ . The resulting mapping $ \partial : \ \pi _{n} ( X ,\ A ,\ x _{0} ) \rightarrow \pi _{n-1} ( A ,\ x _{0} ) $ is a morphism of pointed sets (if $ n > 1 $ a homomorphism of groups) and is called a boundary homomorphism or a boundary operator. The boundary homomorphism, together with the homomorphisms $ i _ \star $ and $ j _ \star $ induced by the imbeddings $ i : \ ( A ,\ x _{0} ) \rightarrow ( X ,\ x _{0} ) $ and $ j : \ ( X ,\ x _{0} ) \rightarrow ( X ,\ A ,\ x _{0} ) $ , makes it possible to write down a sequence of groups and homomorphisms, infinite from the left:$$ {} \dots \stackrel{ {j _{\#}}} \rightarrow \pi _{n+1} ( X ,\ A ,\ x _{0} ) \stackrel \partial \rightarrow \pi _{n} ( A ,\ x _{0} ) \stackrel{ {i _{\#}}} \rightarrow \pi _{n} ( X ,\ x _{0} ) \stackrel{ {j _{\#}}} \rightarrow $$ $$ \stackrel{ {j _{\#}}} \rightarrow \pi _{n} ( A ,\ x _{0} ) \stackrel \partial \rightarrow \dots $$ $$ {} \dots \stackrel{ {j _{\#}}} \rightarrow \pi _{2} ( X ,\ A ,\ x _{0} ) \stackrel \partial \rightarrow \pi _{1} ( A ,\ x _{0} ) \stackrel{ {i _{\#}}} \rightarrow \pi _{1} ( X ,\ x _{0} ) \stackrel{ {j _{\#}}} \rightarrow $$ $$ \stackrel{ {j _{\#}}} \rightarrow \pi _{1} ( X ,\ A ,\ x _{0} ) \stackrel \partial \rightarrow \pi _{0} ( A ,\ x _{0} ) \stackrel{ {i _{\#}}} \rightarrow \pi _{0} ( X ,\ x _{0} ) . $$ This is an exact sequence; it is called the exact homotopy sequence of the pair $ ( X ,\ A ,\ x _{0} ) $ and is usually denoted by $ \pi ( X ,\ A ,\ x _{0} ) $ . If $ \pi _{n} ( X ,\ x _{0} ) = 0 $ for all $ n \geq 0 $ , then the homomorphism $ \partial : \ \pi _{n} ( X ,\ A ,\ x _{0} ) \rightarrow \pi _{n-1} ( A ,\ x _{0} ) $ is an isomorphism (also for all $ n $ ).


The boundary homomorphism $ \partial $ is natural, that is, it is a morphism of the functor $ \pi _{n} $ into the functor $ \pi _{n-1} \circ \iota $ ( more exactly, into the functor $ \pi _{n-1} \circ \iota ^ \prime $ where $ \iota ^ \prime : \ ( X ,\ A ,\ x _{0} ) \mapsto ( A ,\ x _{0} ,\ x _{0} ) $ ). This makes it possible to define $ \pi ( X ,\ A ,\ x _{0} ) $ as a functor that takes values in the category of exact sequences of pointed sets which, except for the last six sets, are Abelian groups and, except for the last three sets, are groups.

Let $ p : \ E \rightarrow B $ be an arbitrary fibration in the sense of Serre and let $ A \subset B $ , $ E ^ \prime = p ^{-1} $ , $ e _{0} \in E ^ \prime $ , and $ b _{0} = p ( e _{0} ) $ . The mapping $ p $ defines a mapping $ p ^ \prime : \ ( E ,\ E ^ \prime ,\ e _{0} ) \rightarrow ( B ,\ A ,\ b _{0} ) $ of pointed pairs. For any $ n \geq 1 $ the induced homomorphism $ p _ \star ^ \prime : \ \pi _{n} ( E ,\ E ^ \prime ,\ e _{0} ) \rightarrow \pi _{n} ( B ,\ A ,\ b _{0} ) $ is an isomorphism. In particular, this is true for $ A = b _{0} $ . In the latter case the formula $ \tau = \partial \circ ( p _ \star ^ \prime ) ^{-1} $ unambiguously defines a homomorphism $ \tau : \ \pi _{n} ( B ,\ b _{0} ) \rightarrow \pi _{n-1} ( F ,\ e _{0} ) $ where $ F = p ^{-1} ( b _{0} ) $ is the fibre of $ p $ over $ b _{0} $ . This homomorphisms is called the homotopy transgression. It occurs in the exact sequence$$ {} \dots \rightarrow \pi _{n} ( F ,\ e _{0} ) \stackrel{ {i _{\#}}} \rightarrow \pi _{n} ( E ,\ e _{0} ) \stackrel{ {p _{\#}}} \rightarrow \pi _{n} ( B ,\ b _{0} ) \stackrel \tau \rightarrow $$ $$ \stackrel \tau \rightarrow \pi _{n-1} ( F ,\ e _{0} ) \rightarrow \dots . $$ This sequence is called the homotopy sequence of the fibration $ p : \ E \rightarrow B $ . Putting a fibration into correspondence with its homotopy sequence yields a functor on the category of all (pointed) fibrations.

In the particular case when $ p $ is the standard Serre fibration of paths over a space $ X $ , for any $ n \geq 0 $ one has the isomorphism $ \pi _{n} ( \Omega X ) \approx \pi _{n+1} (X) $ , where $ \Omega X $ is the loop space of $ X $ . This isomorphism is called the Hurewicz isomorphism.

The above properties actually unambiguously define the homotopy groups $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ , i.e. may be taken as axioms which describe these groups. In fact, let $ \pi _{1} \dots \pi _{n} \dots $ be an arbitrary sequence of homotopy-invariant normalized functors, defined on the category of pointed spaces, taking values in the category of pointed sets, and having the following property: For any fibration in the sense of Serre $ p : \ E \rightarrow B $ , any subset $ A \subset B $ and any point $ e _{0} \in p ^{-1} (A) $ , the induced homomorphism $ \pi _{n} ( E ,\ p ^{-1} (A) ,\ e _{0} ) \rightarrow \pi _{n} ( B ,\ A ,\ p ( e _{0} ) ) $ is an isomorphism. Such a sequence is called a homotopy system if for any $ n \geq 1 $ there is defined a morphism $ \partial $ of the functor $ \pi _{n} $ into the functor $ \pi _{n-1} \circ \iota ^ \prime $ ( if $ n = 1 $ , into $ \pi _{0} ( X ,\ x _{0} ) $ ) that is an isomorphism for any pointed pair $ ( X ,\ A ,\ x _{0} ) $ for which $ \pi _{n} (X ,\ x _{0} ) = 0 $ for all $ n \geq 0 $ . Any homotopy system is isomorphic to the homotopy system constructed above, which consists of homotopy groups. Furthermore, if $ n \geq 3 $ , a group structure can be uniquely introduced into the pointed sets $ \pi _{n} (X,\ A,\ x _{0} ) $ ( and also into the sets $ \pi _{2} ( X ,\ x _{0} ) $ ) so that all morphisms $ f _ \star $ are homomorphism (this structure accordingly corresponds to that described by formula (1)). On the other hand, the sets $ \pi _{2} ( X ,\ A ,\ x _{0} ) $ if $ A \neq x _{0} $ and $ \pi _{1} ( X ,\ x _{0} ) $ carry only the inverse group operation. All this means that the above properties unambiguously define the homotopy groups (up to the order of multiplication in non-commutative groups).

For any mapping $ u : \ ( I ^{n} ,\ \dot{I} ^{n} ) \rightarrow ( X ,\ x _{2} ) $ and any path $ \nu : \ I \rightarrow X $ connecting two points $ x _{1} $ and $ x _{2} $ , the formula $ g _{t} (x) = \nu ( 1 - t ) ,\ x \in \dot{I} ^{n} $ , defines a homotopy of $ u \mid _ {\dot{I} ^{n}} $ . By the homotopy extension axiom (cf. Cofibration) this homotopy can be extended to a homotopy $ u _{t} : \ I ^{n} \rightarrow X $ for which $ u _{0} = u $ . The final mapping $ u _{1} $ of this homotopy maps $ \dot{I} ^{n} $ into $ x _{1} $ , i.e. represents a mapping $ ( I ^{n} ,\ \dot{I} ^{n} ) \rightarrow ( X ,\ x _{1} ) $ . The corresponding element of the homotopy group depends only on the class $ [ u ] \in \pi _{n} ( X ,\ x _{2} ) $ of $ u $ and the homotopy class $ \alpha = [ \nu ] $ of $ \nu $ , and is denoted by the symbol $ \alpha x $ ( if $ n = 1 $ , by the symbol $ x ^ \alpha $ ). The family $ G _{x} = \pi _{n} ( X ,\ x ) $ is thus defined as a local family on the space $ X $ , i.e. on the fundamental groupoid of this space. In particular, for any point $ x _{0} \in X $ the group $ \pi _{1} ( X ,\ x _{0} ) $ operates on $ \pi _{n} ( X ,\ x _{n} ) $ . If $ n = 1 $ these operators act as inner automorphisms: $ x ^ \alpha = \alpha x \alpha ^{-1} $ , and if $ n > 1 $ they make the group $ \pi _{n} ( X ,\ x _{0} ) $ into a $ \pi _{1} ( X ,\ x _{0} ) $ - module. For any continuous mapping $ f : \ ( X ,\ x _{0} ) \rightarrow ( Y ,\ y _{0} ) $ the induced homomorphisms $ f _ \star : \ \pi _{n} ( X ,\ x _{0} ) \rightarrow \pi _{n} ( Y ,\ y _{0} ) $ are operator homomorphisms (homomorphisms of modules): $ f _ \star ( \alpha x ) = f _ \star ( \alpha ) f _ \star (x) $ .


In a similar way, the groups $ G _{x} = \pi _{n} ( X ,\ A ,\ x ) $ , $ x \in A $ , constitute a local family of homotopy groups on the subspace $ A $ . In particular, the group $ \pi _{1} ( A ,\ x _{0} ) $ operates on the homotopy group $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ so that if $ n > 2 $ the group $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ is a $ \pi _{1} ( X ,\ A ,\ x _{0} ) $ - module. The group $ \pi _{2} ( X ,\ A ,\ x _{0} ) $ is said to be a crossed $ ( \pi _{1} ( A ,\ x _{0} ) ,\ \partial ) $ - module (cf. Crossed modules), where $ \partial : \ \pi _{2} ( X ,\ A ,\ x _{0} ) \rightarrow \pi _{1} ( A ,\ x _{0} ) $ is the boundary homomorphism.

The group $ \pi _{1} ( A ,\ x _{0} ) $ acts as a group of operators not only on the groups $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ but also on the groups $ \pi _{n} ( A ,\ x _{0} ) $ , and also, by virtue of the natural homomorphism $ \pi _{1} ( A ,\ x _{0} ) \rightarrow \pi _{1} ( X ,\ x _{0} ) $ , on the groups $ \pi _{n} ( X ,\ x _{0} ) $ . With respect to these actions of $ \pi _{1} ( A ,\ x _{0} ) $ all homomorphisms of the exact sequence $ \pi ( X ,\ A ,\ x _{0} ) $ are operator homomorphisms, so that $ \pi _{1} ( A ,\ x _{0} ) $ can be regarded as a group of operators on the sequence $ \pi ( X ,\ A ,\ x _{0} ) $ . This is equivalent to saying that the sequences $ \pi ( X ,\ A ,\ x ) $ , $ x \in A $ , constitute a local family of exact sequences of the subspace $ A $ .


If the complement $ X \setminus A $ is represented as a union of disjoint open $ n $ - dimensional cells, then the $ \pi _{1} ( A ,\ x _{0} ) $ - module $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ is a free module (if $ n = 2 $ , a free crossed module) and has a system of free generators — a basis in bijective (not necessarily natural) correspondence with the cells of $ X \setminus A $ ( Whitehead's theorem).

The mappings $ ( I ^{n} ,\ \dot{I} ^{n} ) \rightarrow ( X ,\ x _{0} ) $ are in bijective correspondence with the mappings $ ( S ^{n} ,\ s _{0} ) \rightarrow ( X ,\ x _{0} ) $ , where $ S ^{n} $ is an $ n $ - dimensional sphere and $ s _{0} $ is some point on it. For this reason the elements of $ \pi _{n} ( X ,\ x _{0} ) $ can be regarded as the homotopy classes of mappings $ ( S ^{n} ,\ s _{0} ) \rightarrow ( X ,\ x _{0} ) $ . This is also true if $ n = 0 $ . The above identification depends on the selection of some relative homeomorphism $ \phi : \ ( I ^{n} ,\ \dot{I} ^{n} ) \rightarrow ( S ^{n} ,\ s _{0} ) $ . It is common to select and fix the sphere $ S ^{n} $ and the homeomorphism $ \phi $ once and for all. In the original definition of Hurewicz, which is not frequently used nowadays, $ S ^{n} $ was not fixed, while $ \phi $ was given up to a homotopy. Such a specification of $ \phi $ is equivalent to specifying an orientation on $ S ^{n} $ . Thus, according to Hurewicz, the elements of $ \pi _{n} ( X ,\ x _{0} ) $ are pointed homotopy classes of mappings of an oriented $ n $ - dimensional sphere into $ X $ . The set $ [ S ^{n} ,\ X ] $ of non-pointed homotopy classes of mappings $ S ^{n} \rightarrow X $ is in bijective correspondence with the orbits of the action of $ \pi _{1} ( X ,\ x _{0} ) $ on $ \pi _{n} ( X ,\ x _{0} ) $ ( cf. Orbit). If $ \pi _{1} ( X ,\ x _{0} ) = 0 $ ( or, more generally, if $ \pi _{1} ( X ,\ x _{0} ) $ acts trivially on $ \pi _{n} ( X ,\ x _{0} ) $ ), then $ X $ is said to be homotopically $ n $ - simple. In this case $ \pi _{n} ( X ,\ x _{0} ) $ is independent of $ x _{0} $ ( so that the notation $ \pi _{n} ( X ) $ is fully justified). This group is naturally identified with the set $ [ S ^{n} ,\ X ] $ , which, as a consequence, has a group structure. A space that is homotopically $ n $ - simple for all $ n $ is said to be Abelian.

Let $ s _{n} $ be the orientation class of the sphere $ S ^{n} $ and let $ h ( [ f ] ) = f _ \star ( s _{n} ) $ , $ [ f ] \in \pi _{n} ( x ,\ x _{0} ) $ . This defines a homomorphism $ h : \ \pi _{n} ( X ,\ x _{0} ) \rightarrow H _{n} ( X ) $ , the so-called Hurewicz homomorphism. Its kernel contains all elements of the form $ \alpha x - x $ , $ x \in \pi _{n} ( X ,\ x _{0} ) $ , $ \alpha \in \pi _{1} ( X ,\ x _{0} ) $ ( if $ n = 1 $ , all elements of the form $ x ^ \alpha x ^{-1} = \alpha x \alpha ^{-1} x ^{-1} $ , i.e. it contains the commutator $ [ \pi _{1} ,\ \pi _{1} ] $ of $ \pi _{1} ( X ,\ x _{0} ) $ ). Poincaré's classical theorem states that for $ n = 1 $ the kernel of $ h $ coincides with the commutator $ [ \pi _{1} ,\ \pi _{1} ] $ , so that the group $ H _{1} (X) $ is isomorphic to the Abelianization of the fundamental group $ \pi _{1} ( X ,\ x _{0} ) $ . Hurewicz's theorem, which is a generalization of Poincaré's theorem to the case $ n > 1 $ , states that if $ \pi _{i} (X) = 0 $ for $ i < n $ , then the homomorphism $ h : \ \pi _{n} (X) \rightarrow H _{n} (X) $ is an isomorphism (and the homomorphism $ h : \ \pi _{n+1} (X) \rightarrow H _{n+1} (X) $ is an epimorphism).

In a similar way, the elements of $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ can be regarded as (pointed) homotopy classes of mappings $ ( E ,\ S ) \rightarrow ( X ,\ A ) $ , where $ E $ is an (oriented) $ n $ - dimensional ball and $ S $ is its boundary. If the pair $ ( X ,\ A ) $ is homotopically $ n $ - simple (i.e. if $ \pi _{1} ( A ,\ x _{0} ) $ acts trivially on $ \pi _{n} ( X ,\ A ,\ x _{0} ) $ ), then the requirement of pointedness may be dropped in this definition. The formula$$ h ( [ f ] ) = f _ \star ( e _{n} ) , $$ where $ e _{n} $ is the orientation class of the pair $ ( E ,\ S ) $ and $ [ f ] \in \pi _{n} ( X ,\ A ,\ x _{0} ) $ defines the Hurewicz homomorphism$$ h : \ \pi _{n} ( X ,\ A ,\ x _{0} ) \rightarrow H _{n} ( X ,\ A ) . $$ If $ \pi _{1} ( A ,\ x _{0} ) = 0 $ and $ \pi _{n} ( X ,\ A ,\ x _{0} ) = 0 $ for $ i < n $ , this homomorphism is an isomorphism (Hurewicz's theorem for relative groups).

Two principal methods are known for the computation of the homotopy groups of specific spaces: the method of killing spaces (cf. Killing space) and the method of homotopy resolutions (cf. Homotopy type; Postnikov system). The first method is based on the isomorphism $ \pi _{n+1} (X) \approx H _{n+1} ( X ,\ n ) $ , which follows from Hurewicz's theorem and the definition of the killing space $ ( X ,\ n ) $ . This isomorphism reduces the computation of $ \pi _{n+1} (X) $ to the problem of computing the homology groups $ H _{n+1} ( X ,\ n ) $ . The space $ ( X ,\ n ) $ fibres over the space $ ( X ,\ n - 1 ) $ with fibre $ K ( \pi _{n} (X) ,\ n - 1 ) $ , and the homology groups of the space $ K ( \pi ,\ n ) $ are known. Therefore one may try to find the lower homology groups of killing spaces by induction. The problem of computing the homology groups of a fibre space from the homology groups of its base and fibre is still not completely solved in its general formulation (and, obviously, a general satisfactory solution does not exist). However, extensive information on the homology groups of the spaces $ ( X ,\ n ) $ can be extracted from the corresponding Serre spectral sequence. In many cases this information is sufficient for the computation of $ H _{n+1} ( X ,\ n ) \approx \pi _{n+1} (X) $ , at least for some $ n $ . An essential technical simplification of the problem is obtained on the basis of the Serre's theory of classes of Abelian groups and the $ G _{p} $ - approximation derived from it. With this theory it is possible to compute entirely in the cohomology and only for the coefficient groups $ \mathbf Z / p $ . The geometric principles on which this technique is based were first clarified by J.F. Adams and D. Sullivan on the basis of the concept of localization of topological spaces at a given prime number $ p $ .


The second (also inductive) method of computing homotopy groups consists of a stepwise construction of the homotopy resolution of the space $ X $ . Suppose the $ n $ - th term of this resolution is known (e.g. if $ X = S ^{n} $ , then $ X _{n} = K ( \mathbf Z ,\ n ) $ ). The next term must be the fibre space over $ X _{n} $ with fibre $ K ( \pi _{n+1} (X) ,\ n + 1 ) $ ; moreover, the group $ H _{n+1} $ must be isomorphic to the known group $ H _{n+1} (X) $ . This gives (on the basis of the corresponding spectral sequence) definite information on the group $ \pi _{n+1} (X) $ , which, in many cases, makes it possible to compute it completely. For example, for $ X = S ^{n} $ by this method all groups $ \pi _{n+k} ( S ^{n} ) $ , $ k \leq 13 $ , can be found. In its modern form, this method is also based on the concept of localization.

The method of homology resolutions was extended (cf. [4]) to an algorithm that is applicable to any simply-connected finite $ \mathop{\rm CW}\nolimits $ - complex and that gives all its homotopy groups. However, for practical use this algorithm is too complicated.

Since the homotopy theory is completely equivalent to the homotopy theory of simplicial sets, the definition of a homotopy group may be transferred to any (complete) simplicial set. The "combinatorial" definition obtained (due to D. Kan) can easily be extended to an algorithm. However, this algorithm is also too complicated for practical use.

From any of the above methods it is easy to establish that the homotopy groups of a simply-connected space having finitely-generated homology groups, are also finitely generated. The analogous statement for non-simply connected spaces (i.e. its homology groups should be finitely generated as $ \pi _{1} (X) $ - modules) is, in general, not true.

Let $ S $ be the (reduced) suspension functor, and let $ \Omega $ be the loop functor. Since these functors are adjoint, the identity mapping $ S X \rightarrow S X $ defines an imbedding $ X \subset \Omega S X $ , for any $ X $ . Since $ \pi _{n} ( \Omega S X ) \approx \pi _{n+1} ( S X ) $ , this imbedding defines a homomorphism$$ E : \ \pi _{n} ( X ) \rightarrow \pi _{n+1} ( S X ) , $$ which is known as the suspension homomorphism. It coincides with the homomorphism obtained by assigning to an arbitrary (pointed) mapping $ f : \ S ^{n} \rightarrow X $ its suspension $ S f : \ S ^{n+1} \rightarrow S X $ . This homomorphism occurs in an exact sequence:$$ {} \dots \rightarrow \pi _{n} (X) \rightarrow ^ E \pi _{n+1} ( S X ) \rightarrow ^ H \pi _{n} ( \Omega S X ,\ X ) \stackrel \partial \rightarrow $$ $$ \stackrel \partial \rightarrow \pi _{n-1} (X) \rightarrow \dots . $$ This sequence is called the suspension sequence of the space $ X $ . The homomorphism $ H $ in it is a generalization of the classical Hopf invariant.

If $ X $ is a countable CW-complex with one vertex, the space $ \Omega SX $ may be replaced by the infinite reduced product $ X _ \infty $ of the complex $ X $ . This shows that if $ \pi _{i} (X) = 0 $ for $ i \leq m $ , then $ E $ is an isomorphism for all $ n \leq 2m - 1 $ and an epimorphism if $ n = 2m - 1 $ . This theorem is known as Freudenthal's suspension theorem (H. Freudenthal first published the proof for the case $ X = S ^{n} $ , although the theorem was known much earlier.)

Freudenthal's theorem shows that for $ k \leq 2n - 1 $ the group $ \pi _{n+k} (S ^{n} ) $ is independent of $ n $ . It is called the $ k $ - th stable homotopy group of the sphere (cf. also Stable homotopy group). Similar stabilization phenomena occur for the homotopy groups of the orthogonal groups, of the Thom spaces $ \mathop{\rm MSO}\nolimits (n) $ ( cf. Thom space) and in many other cases. The general study of these phenomena is most conveniently done within the framework of the so-called theory of spectra. In this theory stable homotopy groups arise as the homotopy groups of spectra. These groups have an essentially simpler structure than the homotopy groups of a space and their study (and computation) is an easier task. For example, for the computation of these groups one has a special device: the Adams spectral sequence.

Homotopy groups have been generalized in various directions. For example, an attempt was made to replace the spheres by other spaces. Here one may note toroidal homotopy groups, obtained by interpreting the Whitehead product as a commutator. It was also shown that the set of homotopy classes of mappings $ X \rightarrow Y $ admits a group operation which is natural with respect to $ Y $ if and only if $ X $ is a co-$ H $ - space. Homotopy groups with coefficients were obtained by replacing the spheres $ S ^{n} $ by the Moore spaces $ M (G,\ n) $ ( cf. Moore space). This definition of homotopy groups with coefficients was not very successful. A more satisfactory definition (compatible with the general Eckmann–Hilton duality principle) was obtained by replacing the Moore $ M $ - spaces by co-$ M $ - spaces. However, these homotopy groups were not defined for all $ G $ ( e.g. for $ G $ the additive group of real numbers, these groups are not defined).

The question of the construction of homotopy groups in categories other than the category of pointed pairs has been studied in detail. First of all one has to mention the homotopy groups of a triad (cf. Triads, see, e.g., [3]), which were very useful in the study of the homomorphism $ E $ . A very general construction of homotopy groups was proposed in connection with studies on duality. On the basis of the concept of a standard construction (see [6]) the construction of homotopy groups was transferred to arbitrary categories. A fundamental role in this construction is played by the homotopy groups of simplicial sets mentioned earlier.

References

[1] N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)
[2] V.G. Boltyanskii, "The homotopy theory of continuous mapping and vector fields" , Moscow (1955) (In Russian)
[3] S.-T. Hu, "Homotopy theory" , Acad. Press (1959)
[4] E.H. Brown, "Finite computability of Postnikov complexes" Ann. of Math. (2) , 65 (1957) pp. 1–20
[5] D. Kan, "A combinatorial definition of homotopy groups" Ann. of Math. (2) , 67 (1958) pp. 282–313
[6] J. Stallings, "A finitely presented group whose 3-dimensional integral homology is not finitely generated" Amer. J. Math. , 85 (1963) pp. 541–543
[7a] B. Eckmann, P. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" C.R. Acad. Sci. , 246 (1958) pp. 2444–2447
[7b] B. Eckmann, P. Hilton, "Groupes d'homotopie et dualité. Suites exactes" C.R. Acad. Sci. , 246 (1958) pp. 2555–2558
[7c] B. Eckmann, P. Hilton, "Groupes d'homotopie et dualité. Coefficients" C.R. Acad. Sci. , 246 (1958) pp. 2991–2993
[7d] B. Eckmann, P. Hilton, "Transgression homotopique et cohomologique" C.R. Acad. Sci. , 247 (1958) pp. 620–623
[7e] B. Eckmann, P. Hilton, "Décomposition homologique d'une polyèdre simplement connexe" C.R. Acad. Sci. , 248 (1959) pp. 2054–2056
[8] D. Sullivan, "Geometric topology" , M.I.T. (1971) (Notes)


Comments

Although W. Hurewicz

was the first to study the higher homotopy groups in detail, the definition was in fact suggested a few years earlier by E. Čech [a2]. The action of the fundamental group on the higher homotopy groups was first studied by S. Eilenberg [a3]. A good general reference for homotopy groups is [a4].

The stable homotopy groups form a generalized homology theory, i.e. a theory which satisfies all the Eilenberg–Steenrod axioms except possibly the dimension axiom. This theory is in fact defined by the spectrum of spheres $ ( S ^{n} ) _{n} $ , cf. Spectrum of spaces. The corresponding generalized cohomology theory (cf. Generalized cohomology theories) defined by this spectrum consists of the cohomotopy group. Cf., e.g., [a4] and [a11] for more details. Powerful tools for computing the stable homotopy groups of the spheres (besides the (classical) Adams spectral sequence) involve the Adams–Novikov spectral sequence, the so-called chromatic spectral sequence and complex cobordism, cf. [a12].

References

[a1a] W. Hurewicz, "Beiträge zur Topologie der Deformationen I-II" Proc. Ned. Akad. Weten. Ser. A , 38 (1935) pp. 112–119; 521–528
[a1b] W. Hurewicz, "Beiträge zur Topologie der Deformationen III-IV" Proc. Ned. Akad. Weten. Ser. A , 39 (1936) pp. 117–126; 215–224
[a2] E. Čech, "Höherdimensionale Homotopiegruppen" , Verh. Intern. Mathematikerkongress Zürich, 1932 , O. Füssli (1932) pp. 203
[a3] S. Eilenberg, "On the relation between the fundamental group of a space and the higher homotopy groups" Fund. Math. , 32 (1939) pp. 167–175
[a4] G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 23; 415–455
[a5] B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. §12
[a6] P.J. Hilton, "An introduction to homotopy theory" , Cambridge Univ. Press (1953)
[a7] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1960)
[a8] D. Sullivan, "Genetics of homotopy theory and the Adams conjecture" Ann. of Math. , 100 (1974) pp. 1–79
[a9] D.G. Quillen, "Homotopical algebra" , Springer (1967)
[a10] B. Eckmann, "Homotopie et dualité" , Coll. Topol. Algébrique Louvain, 1956 , Masson (1957) pp. 41–53
[a11] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975)
[a12] D.C. Ravenel, "Complex cobordism and stable homotopy groups of spheres" , Acad. Press (1986)
How to Cite This Entry:
Homotopy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homotopy_group&oldid=44244
This article was adapted from an original article by M.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article